TITLE:
Minimizing the Variance of a Weighted Average
AUTHORS:
Doron J. Shahar
KEYWORDS:
Variance, Weighted Average, Minimization
JOURNAL NAME:
Open Journal of Statistics,
Vol.7 No.2,
April
24,
2017
ABSTRACT: It is common practice in science to take a weighted average of estimators of a single parameter. If the original estimators are unbiased, any weighted average will be an unbiased estimator as well. The best estimator among the weighted averages can be obtained by choosing weights that minimize the variance of the weighted average. If the variances of the individual estimators are given, the ideal weights have long been known to be the inverse of the variance. Nonetheless, I have not found a formal proof of this result in the literature. In this article, I provide three different proofs of the ideal weights.